The name of the course is Introduction to Open Data Science and we are focusing to language R, RStudio, GitHub and Markdown. You can find my Github repository here.
After the data wrangling exercise the new data set is found from the data folder. The set is based on data that was collected from course Introduction to Social Statistics, fall 2014 - in Finnish. The survey was conducted 3.12.2014 - 10.1.2015 by Kimmo Vehkalahti.
student2014 <- read.table("data/learning2014.txt", header = TRUE)
dim(student2014)
## [1] 166 7
The student data includes 7 variables and 166 rows.
str(student2014)
## 'data.frame': 166 obs. of 7 variables:
## $ gender : Factor w/ 2 levels "F","M": 1 2 1 2 2 1 2 1 2 1 ...
## $ age : int 53 55 49 53 49 38 50 37 37 42 ...
## $ attitude: int 37 31 25 35 37 38 35 29 38 21 ...
## $ deep : num 3.58 2.92 3.5 3.5 3.67 ...
## $ stra : num 3.38 2.75 3.62 3.12 3.62 ...
## $ surf : num 2.58 3.17 2.25 2.25 2.83 ...
## $ points : int 25 12 24 10 22 21 21 31 24 26 ...
Variables deep, strat and surf are combination variables from the original survey data.
summary(student2014)
## gender age attitude deep stra
## F:110 Min. :17.00 Min. :14.00 Min. :1.583 Min. :1.250
## M: 56 1st Qu.:21.00 1st Qu.:26.00 1st Qu.:3.333 1st Qu.:2.625
## Median :22.00 Median :32.00 Median :3.667 Median :3.188
## Mean :25.51 Mean :31.43 Mean :3.680 Mean :3.121
## 3rd Qu.:27.00 3rd Qu.:37.00 3rd Qu.:4.083 3rd Qu.:3.625
## Max. :55.00 Max. :50.00 Max. :4.917 Max. :5.000
## surf points
## Min. :1.583 Min. : 7.00
## 1st Qu.:2.417 1st Qu.:19.00
## Median :2.833 Median :23.00
## Mean :2.787 Mean :22.72
## 3rd Qu.:3.167 3rd Qu.:27.75
## Max. :4.333 Max. :33.00
From 166 survey respondents 56 was men and 110 was females. The mean of age was 25,5 years. Oldes respondet was 55 and youngest 17 years old.
Variables differ between genders. Distributions are different in age, attitude and surf (surfface). Three highest correlation between variables are:
##
## Call:
## lm(formula = points ~ attitude + stra + surf, data = student2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -17.1550 -3.4346 0.5156 3.6401 10.8952
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.01711 3.68375 2.991 0.00322 **
## attitude 0.33952 0.05741 5.913 1.93e-08 ***
## stra 0.85313 0.54159 1.575 0.11716
## surf -0.58607 0.80138 -0.731 0.46563
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.296 on 162 degrees of freedom
## Multiple R-squared: 0.2074, Adjusted R-squared: 0.1927
## F-statistic: 14.13 on 3 and 162 DF, p-value: 3.156e-08
In this linear regression model points are the target variable and attitude, strategy (stra) and surfface (surf) are explanatory variables.
Residuals of the model are between ~ -17.2 and ~10.9 when median is 0.52. I assume that errors are normally distributed but distribution needs confirmation.
Attitude is the only variable that has a very good significance in this model. p-value of stra and surf is too high to be even slightly significance.
Variables estimated coefficient is ~0.34 and it’s standard error is clearly smaller (~0.057). Other explanatory variables have not significance in this model.
Residual standard error is high in relation to a first and third quantiles of residuals.
This linear regression model of three explanatory variables explains ~19.3% of the points.
##
## Call:
## lm(formula = points ~ attitude, data = student2014)
##
## Residuals:
## Min 1Q Median 3Q Max
## -16.9763 -3.2119 0.4339 4.1534 10.6645
##
## Coefficients:
## Estimate Std. Error t value Pr(>|t|)
## (Intercept) 11.63715 1.83035 6.358 1.95e-09 ***
## attitude 0.35255 0.05674 6.214 4.12e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 5.32 on 164 degrees of freedom
## Multiple R-squared: 0.1906, Adjusted R-squared: 0.1856
## F-statistic: 38.61 on 1 and 164 DF, p-value: 4.119e-09
Estimated coefficient of explanatory variable attitude is ~0.35. This means that when attitude rises one point target variable (points) grows 0.35 times.
Attitude p-value (0.00000000412) shows that variable is very significant in this linear regression model
This model explains 18.6% of the points (target variable)
Residuals vs Fitted This plot shows that there is no pattern between residuals. There is a constant variance among errors. One can confirm that the assumption of constant variance of errors is valid.
Normal Q_Q Normal QQ-plot shows that the errors of the model are normally distributed.
Residual vs Leverage Last plot of the graphical model validation shows that the impact of the singel observation is moderate. Model includes some outliers but the leverage of singel observation don’t compromise the validation of the model.
Using Data Mining To Predict Secondary School Student Alcohol Consumption. Fabio Pagnotta, Hossain Mohammad Amran Department of Computer Science,University of Camerino
https://archive.ics.uci.edu/ml/datasets/STUDENT+ALCOHOL+CONSUMPTION
Data is rolled into one from Math course and Portuguese language course datasets. After the data wrangling exercise the new data set is found from the data folder.
alc <- read.table("data/student_alc.txt", header = TRUE)
dim(alc)
## [1] 382 35
The student data includes 35 variables and 382 rows.
In this analysis I’m going to study the relationship of high/low alcohol consumption between sex and the following variables:
| Variable | Type | Description |
|---|---|---|
| age | numeric | student’s age |
| studytime | numeric, scale 1-4 | weekly study time |
| freentime | numeric, scale 1-5 | free time after school |
| absence | numeric | number of school absences |
First these relationships are observed from tables and graphics. Hypothesis are as follows:
Age
The age of high consumption of alcohol may differ between sex. The development of charcter differs between young people and this may affect on habbits of alcohol consumption.
H0: Age don’t affect on alcohol consumption
H1: There is difference in between level of alcohol consumption and age
#a jitterplot of high_use, sex and age
g1 <- ggplot(alc, aes(x = high_use, y = age, col = sex))
g1 + geom_jitter() + ggtitle("Age by alcohol consumption and sex")
It seems that there is randomnes of sex and age in both alcohol consumption groups.
#a boxplot of high_use, sex and age
g1 <- ggplot(alc, aes(x = high_use, y = age, col = sex))
g1 + geom_boxplot() + ggtitle("Age by alcohol consumption and sex") + xlab("High consumption group") + ylab("Age of student")
Means however show that young male students have lower mean of age in low consumption group and females in high consumption group.
Study time
High alcohol consumption may be related to time spend in studies because one can’t do both at the same time at least not successfully.
H0: Alcohol consumption do not affect on weekly study time
H1: Alcohol consumption affects on weekly study time (numeric: 1 - <2 hours, 2 - 2 to 5 hours, 3 - 5 to 10 hours, or 4 - >10 hours)
#barplot about study time
ggplot(alc, aes(studytime, fill = high_use)) + geom_bar(position = "fill") +
ggtitle("Barplot about study time grouped by high_use")+ xlab("Study time") + ylab("Propotions of students") + scale_y_continuous(name = waiver(), breaks = waiver(), minor_breaks = waiver(), labels = waiver(), limits = NULL, expand = waiver(), na.value = NA_real_, trans = "identity")
It seems that there is less high users in those students groups who spend more time in studying (3-4) than in those who spend less time in studying (1-2).
Free time
High alcohol consumption may be related to free time so that studets who have more free time are consuming more alcohol that studets who haven’t as much free time.
H0: Amount of free time do not affect on alcohol consumption among students
H1: Amount of free time affects on alcohol consumption among students
#barplot about free time
ggplot(alc, aes(freetime, fill = high_use)) + geom_bar(position = "fill") +
ggtitle("Barplot about free time grouped by high_use")+ xlab("Free time") + ylab("Propotions of students") + scale_y_continuous(name = waiver(), breaks = waiver(), minor_breaks = waiver(), labels = waiver(), limits = NULL, expand = waiver(), na.value = NA_real_, trans = "identity")
It seems that there are more students that are consuming alcohol high amounts in those students groups that have more free time than in those who have not as much free time.
Absences
High consumption of alcohol may cause absences.
H0: High consumption of alcohol do not affect on absences
H1: High consumption of alcohol does affect on absences
#a boxplot of high_use and absences
g1 <- ggplot(alc, aes(x = high_use, y = absences, fill = high_use))
g1 + geom_boxplot() + ggtitle("Absences by alcohol consumption") + xlab("High consumption group") + ylab("Absences")
It seems that the differences between alcohol consumption groups in absences are small. Mean of absences is higher in high consumption group but it may not be significant.
#the model with glm()
m <- glm(high_use ~ age + studytime + freetime + absences, data = alc, family = "binomial")
summary(m)
##
## Call:
## glm(formula = high_use ~ age + studytime + freetime + absences,
## family = "binomial", data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.9649 -0.8267 -0.6238 1.0990 2.2861
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.39240 1.79306 -2.450 0.014299 *
## age 0.18543 0.10313 1.798 0.072183 .
## studytime -0.51842 0.15867 -3.267 0.001086 **
## freetime 0.33015 0.12430 2.656 0.007907 **
## absences 0.07856 0.02269 3.463 0.000535 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 465.68 on 381 degrees of freedom
## Residual deviance: 423.18 on 377 degrees of freedom
## AIC: 433.18
##
## Number of Fisher Scoring iterations: 4
From the fitted model one can see that all explanatory variables except age are statistically significant with p-value < 0,01. Variable absence is also statistically sisgnificant with p-value < 0,001. It seems that age doesn’t explain whether or not a student is a high user of alcohol.
# compute odds ratios (OR)
OR <- coef(m) %>% exp
# compute confidence intervals (CI)
CI <- confint(m) %>% exp
## Waiting for profiling to be done...
# print out the odds ratios with their confidence intervals
cbind(OR, CI)
## OR 2.5 % 97.5 %
## (Intercept) 0.01237102 0.0003481843 0.3998333
## age 1.20373817 0.9852330457 1.4775972
## studytime 0.59545916 0.4321414645 0.8061535
## freetime 1.39118359 1.0938389338 1.7826010
## absences 1.08172440 1.0368164005 1.1336048
Odds ratio (OR) and the 95% confidence interval (CI) shows that those students who have a low study time are almost two times as likely to be a high user of alcohol than those studets who have higher study time. Students that have more freetime are also more likely to be a high user of alcohol. Also absences are positively correlated with high use of alcohol. Confidence interval shows that age is not statistically significant (because the interval contains 1) and other variables are.
Predictive power of the final logistic regression model is calculated without the statistically insignificant variable age.
#the model with glm() and without the age variable
m_final <- glm(high_use ~ studytime + freetime + absences, data = alc, family = "binomial")
summary(m_final)
##
## Call:
## glm(formula = high_use ~ studytime + freetime + absences, family = "binomial",
## data = alc)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -1.9420 -0.8332 -0.6450 1.1266 2.1537
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -1.34105 0.56337 -2.380 0.017293 *
## studytime -0.50496 0.15691 -3.218 0.001290 **
## freetime 0.32626 0.12379 2.636 0.008401 **
## absences 0.08324 0.02262 3.680 0.000233 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for binomial family taken to be 1)
##
## Null deviance: 465.68 on 381 degrees of freedom
## Residual deviance: 426.47 on 378 degrees of freedom
## AIC: 434.47
##
## Number of Fisher Scoring iterations: 4
#predict and add the answer and the prediction to the data (alc)
probabilities <- predict(m_final, type = "response")
alc <- mutate(alc, probability = probabilities)
alc <- mutate(alc, prediction = probabilities > 0.5)
#tabulate the target variable versus the prediction
table("High use" = alc$high_use, "Prediction" = alc$prediction)
## Prediction
## High use FALSE TRUE
## FALSE 254 14
## TRUE 93 21
Table shows that the model predict 254 true negatives, 21 true positives, 14 false negatives and 93 false postives. This is sometimes called “confusion table”
table("High use" = alc$high_use, "Prediction" = alc$prediction) %>% prop.table() %>% addmargins()
## Prediction
## High use FALSE TRUE Sum
## FALSE 0.66492147 0.03664921 0.70157068
## TRUE 0.24345550 0.05497382 0.29842932
## Sum 0.90837696 0.09162304 1.00000000
Propabilities of the same table shows that 90,8% is predicted to be false but only 66,5% of them is correct. 9,2% is predicted to be true but 5,5% of them realy are students with high use of alcohol.
#a plot of 'high_use' versus 'probability' in 'alc'
g <- ggplot(alc, aes(x = probability, y = high_use, col = prediction))
g + geom_point()
#defining a loss function (mean prediction error)
loss_func <- function(class, prob) {
n_wrong <- abs(class - prob) > 0.5
mean(n_wrong)
}
#calling loss_func to compute the average number of wrong predictions in the (training) data
loss_func(class = alc$high_use, prob = alc$probability)
## [1] 0.2801047
The average number of wrong predictions in trainig data is 28%.
#computing the average number of wrong predictions in the (training) data
#loss_func(class = alc$high_use, prob = alc$probability)
#K-fold cross-validation
library(boot)
cv <- cv.glm(data = alc, cost = loss_func, glmfit = m_final, K = 10)
#average number of wrong predictions in the cross validation
cv$delta[1]
## [1] 0.2774869
10-fold cross-validation gives good estimate of the actual predictive power of the model. Low value = good.
In this exercise we use Boston data from MASS-library. This dataset contains information collected by the U.S Census Service concerning housing in the area of Boston Mass. Data includes 14 variables and 506 rows.
## [1] 506 14
## 'data.frame': 506 obs. of 14 variables:
## $ crim : num 0.00632 0.02731 0.02729 0.03237 0.06905 ...
## $ zn : num 18 0 0 0 0 0 12.5 12.5 12.5 12.5 ...
## $ indus : num 2.31 7.07 7.07 2.18 2.18 2.18 7.87 7.87 7.87 7.87 ...
## $ chas : int 0 0 0 0 0 0 0 0 0 0 ...
## $ nox : num 0.538 0.469 0.469 0.458 0.458 0.458 0.524 0.524 0.524 0.524 ...
## $ rm : num 6.58 6.42 7.18 7 7.15 ...
## $ age : num 65.2 78.9 61.1 45.8 54.2 58.7 66.6 96.1 100 85.9 ...
## $ dis : num 4.09 4.97 4.97 6.06 6.06 ...
## $ rad : int 1 2 2 3 3 3 5 5 5 5 ...
## $ tax : num 296 242 242 222 222 222 311 311 311 311 ...
## $ ptratio: num 15.3 17.8 17.8 18.7 18.7 18.7 15.2 15.2 15.2 15.2 ...
## $ black : num 397 397 393 395 397 ...
## $ lstat : num 4.98 9.14 4.03 2.94 5.33 ...
## $ medv : num 24 21.6 34.7 33.4 36.2 28.7 22.9 27.1 16.5 18.9 ...
| variable | description |
|---|---|
| crim | per capita crime rate by town |
| zn | proportion of residential land zoned for lots over 25,000 sq.ft. |
| indus | proportion of non-retail business acres per town |
| chas | Charles River dummy variable (= 1 if tract bounds river; 0 otherwise) |
| nox | nitrogen oxides concentration (parts per 10 million) |
| rm | average number of rooms per dwelling |
| age | proportion of owner-occupied units built prior to 1940 |
| dis | weighted mean of distances to five Boston employment centres |
| rad | index of accessibility to radial highways |
| tax | full-value property-tax rate per $10,000 |
| ptratio | pupil-teacher ratio by town |
| black | 1000(Bk - 0.63)^2 where Bk is the proportion of blacks by town |
| lstat | lower status of the population (percent) |
| medv | median value of owner-occupied homes in $1000 |
There are some very intresting distributions fo variables in the plot matrix. Variable rad has high and low values so the plot shows that the values are consenrated either side of the plot. VAriable *
Plotted correlation matrix shows that there is some high correlation between variables:
Correlation is quite clear between industrial areas (indus) and nitrogen oxides (nox). Industry adds pollution in the area. Industry seems to correlate also with variablrs like age, dis, ras and tax.
Nitrogen oxides (nox) correlations are very similar with industry (indus)
Crime rate (crim) seems to correlate with good accessibilitty to radial highways (rad) and value property (tax).
Old houses (age) and employment centers have also something common
summary(Boston)
## crim zn indus chas
## Min. : 0.00632 Min. : 0.00 Min. : 0.46 Min. :0.00000
## 1st Qu.: 0.08204 1st Qu.: 0.00 1st Qu.: 5.19 1st Qu.:0.00000
## Median : 0.25651 Median : 0.00 Median : 9.69 Median :0.00000
## Mean : 3.61352 Mean : 11.36 Mean :11.14 Mean :0.06917
## 3rd Qu.: 3.67708 3rd Qu.: 12.50 3rd Qu.:18.10 3rd Qu.:0.00000
## Max. :88.97620 Max. :100.00 Max. :27.74 Max. :1.00000
## nox rm age dis
## Min. :0.3850 Min. :3.561 Min. : 2.90 Min. : 1.130
## 1st Qu.:0.4490 1st Qu.:5.886 1st Qu.: 45.02 1st Qu.: 2.100
## Median :0.5380 Median :6.208 Median : 77.50 Median : 3.207
## Mean :0.5547 Mean :6.285 Mean : 68.57 Mean : 3.795
## 3rd Qu.:0.6240 3rd Qu.:6.623 3rd Qu.: 94.08 3rd Qu.: 5.188
## Max. :0.8710 Max. :8.780 Max. :100.00 Max. :12.127
## rad tax ptratio black
## Min. : 1.000 Min. :187.0 Min. :12.60 Min. : 0.32
## 1st Qu.: 4.000 1st Qu.:279.0 1st Qu.:17.40 1st Qu.:375.38
## Median : 5.000 Median :330.0 Median :19.05 Median :391.44
## Mean : 9.549 Mean :408.2 Mean :18.46 Mean :356.67
## 3rd Qu.:24.000 3rd Qu.:666.0 3rd Qu.:20.20 3rd Qu.:396.23
## Max. :24.000 Max. :711.0 Max. :22.00 Max. :396.90
## lstat medv
## Min. : 1.73 Min. : 5.00
## 1st Qu.: 6.95 1st Qu.:17.02
## Median :11.36 Median :21.20
## Mean :12.65 Mean :22.53
## 3rd Qu.:16.95 3rd Qu.:25.00
## Max. :37.97 Max. :50.00
All the variables are numerical so we can use scale()-function to scale whole data set.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865
## [1] "matrix"
Scaling the data makes variables look as if they are in the same range. Variables like black and tax were before scaling hundred fold compared to some other variables.
Variable crim is the base of the new categorical variable crime.
| categories | quantile points |
|---|---|
| low | 0%-25% |
| med_low | 25%-50% |
| med_high | 50%-75% |
| high | 75%-100% |
Quantile points of the variable crim
## 0% 25% 50% 75% 100%
## -0.419366929 -0.410563278 -0.390280295 0.007389247 9.924109610
## crime
## low med_low med_high high
## 127 126 126 127
## zn indus chas nox
## Min. :-0.48724 Min. :-1.5563 Min. :-0.2723 Min. :-1.4644
## 1st Qu.:-0.48724 1st Qu.:-0.8668 1st Qu.:-0.2723 1st Qu.:-0.9121
## Median :-0.48724 Median :-0.2109 Median :-0.2723 Median :-0.1441
## Mean : 0.00000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.04872 3rd Qu.: 1.0150 3rd Qu.:-0.2723 3rd Qu.: 0.5981
## Max. : 3.80047 Max. : 2.4202 Max. : 3.6648 Max. : 2.7296
## rm age dis rad
## Min. :-3.8764 Min. :-2.3331 Min. :-1.2658 Min. :-0.9819
## 1st Qu.:-0.5681 1st Qu.:-0.8366 1st Qu.:-0.8049 1st Qu.:-0.6373
## Median :-0.1084 Median : 0.3171 Median :-0.2790 Median :-0.5225
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4823 3rd Qu.: 0.9059 3rd Qu.: 0.6617 3rd Qu.: 1.6596
## Max. : 3.5515 Max. : 1.1164 Max. : 3.9566 Max. : 1.6596
## tax ptratio black lstat
## Min. :-1.3127 Min. :-2.7047 Min. :-3.9033 Min. :-1.5296
## 1st Qu.:-0.7668 1st Qu.:-0.4876 1st Qu.: 0.2049 1st Qu.:-0.7986
## Median :-0.4642 Median : 0.2746 Median : 0.3808 Median :-0.1811
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 1.5294 3rd Qu.: 0.8058 3rd Qu.: 0.4332 3rd Qu.: 0.6024
## Max. : 1.7964 Max. : 1.6372 Max. : 0.4406 Max. : 3.5453
## medv crime
## Min. :-1.9063 low :127
## 1st Qu.:-0.5989 med_low :126
## Median :-0.1449 med_high:126
## Mean : 0.0000 high :127
## 3rd Qu.: 0.2683
## Max. : 2.9865
Training set contains 80% of the data. 20% is in the test set.
## [1] 186 322 180 417 485 280 379 81 168 103 247 499 391 116 20 502 320
## [18] 427 355 178 245 413 47 6 197 67 97 396 269 130 403 104 136 315
## [35] 284 412 241 430 72 62 100 123 258 167 375 52 327 240 101 364 236
## [52] 331 195 234 279 432 409 411 345 45 43 21 368 131 382 233 108 125
## [69] 99 446 264 281 303 344 129 333 10 348 297 451 156 51 83 25 22
## [86] 468 469 250 13 455 183 203 490 217 349 360 26 481 302 329 124 133
## [103] 314 299 254 93 387 201 50 270 15 184 450 211 448 480 321 121 418
## [120] 335 259 82 188 248 63 138 429 489 137 500 179 208 230 443 87 504
## [137] 416 228 363 251 486 440 330 164 414 206 398 86 339 185 341 143 479
## [154] 224 95 290 135 161 255 300 456 126 505 239 96 421 488 393 193 397
## [171] 165 476 464 216 316 38 177 214 420 374 65 115 497 118 166 110 252
## [188] 160 503 328 210 438 337 353 453 493 127 260 286 401 380 447 466 332
## [205] 189 105 78 305 154 225 190 482 266 484 212 60 242 227 122 243 491
## [222] 147 73 483 113 383 437 76 276 46 237 268 288 257 318 467 49 338
## [239] 428 31 61 146 326 296 176 498 24 287 202 200 307 191 37 94 173
## [256] 253 370 42 340 235 75 1 192 68 159 155 462 207 244 439 34 16
## [273] 319 39 77 35 132 238 436 170 285 205 277 419 140 282 148 372 334
## [290] 354 218 71 92 367 442 325 487 199 474 312 219 204 182 53 465 475
## [307] 28 57 9 54 7 232 66 220 323 142 8 386 405 151 471 388 80
## [324] 392 272 152 400 317 229 162 308 452 385 457 44 18 292 256 149 112
## [341] 313 449 271 306 358 32 362 107 463 106 458 14 2 441 114 431 501
## [358] 359 425 445 310 389 29 477 150 373 134 301 181 394 90 492 460 336
## [375] 495 85 246 58 311 384 406 249 294 119 23 304 175 293 215 295 261
## [392] 12 342 36 369 102 172 291 444 346 283 265 198 365
First the linear discriminant analysis (LDA) is fitted to the train set. The new categorical variable crime is the target variable and all the other variables of the dataset are predictor variables.
After fitting we draw the LDA biplot with arrows.
## Call:
## lda(crime ~ ., data = train)
##
## Prior probabilities of groups:
## low med_low med_high high
## 0.2549505 0.2574257 0.2500000 0.2376238
##
## Group means:
## zn indus chas nox rm
## low 0.98889890 -0.8846501 -0.15765625 -0.8687426 0.40827464
## med_low -0.09969684 -0.2674719 -0.08304540 -0.5643705 -0.14716922
## med_high -0.38365582 0.2003265 0.11748284 0.3809752 0.08160335
## high -0.48724019 1.0149946 -0.06727176 1.0521399 -0.41711463
## age dis rad tax ptratio black
## low -0.8989016 0.8521780 -0.6741266 -0.7486137 -0.5135669 0.3770142
## med_low -0.3258397 0.3374276 -0.5456746 -0.4736838 -0.1082613 0.3094517
## med_high 0.3834406 -0.3572742 -0.4144603 -0.3025424 -0.2305360 0.1111525
## high 0.7955508 -0.8361604 1.6596029 1.5294129 0.8057784 -0.8466439
## lstat medv
## low -0.74806739 0.506935581
## med_low -0.14471908 0.001451274
## med_high 0.01121753 0.136705280
## high 0.93200127 -0.699891284
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## zn 0.13853250 0.8046371288 -0.96451679
## indus 0.04391570 -0.1680047542 0.26465188
## chas -0.08371156 -0.0372860741 0.08552400
## nox 0.22055079 -0.7841836940 -1.54242664
## rm -0.10710158 -0.1299699406 -0.17553623
## age 0.25615162 -0.2961059881 0.02677354
## dis -0.12284019 -0.3566217762 0.20012963
## rad 3.44990822 1.0867302850 -0.05182276
## tax -0.01093013 -0.2499253231 0.66756378
## ptratio 0.12045867 0.0153931459 -0.38730853
## black -0.14829277 -0.0007841421 0.07953713
## lstat 0.22648702 -0.2292112397 0.27745623
## medv 0.18076295 -0.3947858497 -0.23971095
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.9513 0.0359 0.0128
## [1] 1 2 1 4 3 2 4 1 3 2 3 2 4 2 3 1 3 4 1 1 2 4 2 1 1 1 2 4 3 3 4 2 3 3 1
## [36] 4 2 4 2 2 1 2 3 3 4 1 3 2 2 4 3 1 1 3 1 4 4 4 1 2 2 3 4 3 4 3 2 2 1 4
## [71] 3 1 2 1 3 1 2 1 1 4 3 2 1 3 3 4 4 2 2 4 2 1 2 1 1 4 3 4 1 1 2 3 3 1 3
## [106] 1 4 1 2 2 3 2 4 2 4 4 2 1 4 1 3 1 1 2 2 3 4 2 3 2 1 2 3 4 1 1 4 3 4 2
## [141] 3 4 1 3 4 2 4 1 1 2 1 3 4 3 1 1 3 3 1 1 4 2 2 2 2 4 4 4 2 4 3 4 4 2 2
## [176] 1 1 2 4 4 1 2 3 2 3 3 2 3 1 2 3 4 1 1 4 2 3 3 1 4 4 4 3 1 2 2 2 1 3 3
## [211] 2 4 3 3 3 2 2 3 1 2 2 3 2 4 2 4 4 2 2 2 3 3 1 1 2 4 2 1 4 3 2 3 2 2 1
## [246] 3 3 1 1 1 1 2 2 1 2 2 4 2 1 3 1 1 1 1 3 3 4 2 2 4 3 3 3 2 2 3 3 3 4 3
## [281] 1 1 2 4 3 1 3 4 1 1 1 2 1 4 4 3 4 1 4 3 2 1 1 1 4 4 3 1 2 1 2 3 1 2 3
## [316] 3 2 4 4 3 4 4 2 4 2 3 4 3 3 3 1 4 4 4 2 3 1 1 3 2 3 4 3 1 4 3 4 2 4 2
## [351] 4 3 1 4 2 4 2 4 4 4 3 4 3 4 3 4 3 1 1 4 1 2 4 1 3 1 2 1 3 4 4 2 2 2 3
## [386] 2 2 1 3 1 3 2 1 1 4 2 3 1 4 1 1 3 1 3
## predicted
## correct low med_low med_high high
## low 14 9 1 0
## med_low 4 14 4 0
## med_high 1 8 15 1
## high 0 0 1 30
Prediction were quite good. There was some errors in the middle of the range but classes low and especially high were good. Only one correct representative of high class was predicted to med_low class.
I’m going to calculate what is the optimal number of clusters for Boston data. First I reload and scale the data. Variables need to be scaled to get comparable distances between observation.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989
## Median : 0.3808 Median :-0.1811 Median :-0.1449
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865
Next I calculate the distances between observations and determinen the number of clusters.
One way to determine the number of clusters is to look how the total of within cluster sum of squares (WCSS) behaves when the number of clusters changes. WCSS was calculated from 1 to 15 clusters. The optimal number of clusters is when the total WCSS drops radically. It seems that in this case optimal number of clusters is two. However we are here to learn so I decided to analyse model with four clusters.
After determining the number of clusters I run the K-means alcorithm again.
It seems that when the data is divided to four clusters there is some clear differences in distriputions of several variables. Crim, zn, indus and blacks are variables were one can distinguish clear patterns between clusters. Some variables (rad & tax) show that sometimes 1 or 2 clusters make a clear distripution but observation of other two clusters are ambigious and there is no clear pattern to be regognised.
After loading the Boston dataset I scale it to get comparable distances.
## crim zn indus
## Min. :-0.419367 Min. :-0.48724 Min. :-1.5563
## 1st Qu.:-0.410563 1st Qu.:-0.48724 1st Qu.:-0.8668
## Median :-0.390280 Median :-0.48724 Median :-0.2109
## Mean : 0.000000 Mean : 0.00000 Mean : 0.0000
## 3rd Qu.: 0.007389 3rd Qu.: 0.04872 3rd Qu.: 1.0150
## Max. : 9.924110 Max. : 3.80047 Max. : 2.4202
## chas nox rm age
## Min. :-0.2723 Min. :-1.4644 Min. :-3.8764 Min. :-2.3331
## 1st Qu.:-0.2723 1st Qu.:-0.9121 1st Qu.:-0.5681 1st Qu.:-0.8366
## Median :-0.2723 Median :-0.1441 Median :-0.1084 Median : 0.3171
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.:-0.2723 3rd Qu.: 0.5981 3rd Qu.: 0.4823 3rd Qu.: 0.9059
## Max. : 3.6648 Max. : 2.7296 Max. : 3.5515 Max. : 1.1164
## dis rad tax ptratio
## Min. :-1.2658 Min. :-0.9819 Min. :-1.3127 Min. :-2.7047
## 1st Qu.:-0.8049 1st Qu.:-0.6373 1st Qu.:-0.7668 1st Qu.:-0.4876
## Median :-0.2790 Median :-0.5225 Median :-0.4642 Median : 0.2746
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6617 3rd Qu.: 1.6596 3rd Qu.: 1.5294 3rd Qu.: 0.8058
## Max. : 3.9566 Max. : 1.6596 Max. : 1.7964 Max. : 1.6372
## black lstat medv clust
## Min. :-3.9033 Min. :-1.5296 Min. :-1.9063 Min. :1.000
## 1st Qu.: 0.2049 1st Qu.:-0.7986 1st Qu.:-0.5989 1st Qu.:2.000
## Median : 0.3808 Median :-0.1811 Median :-0.1449 Median :3.000
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean :2.674
## 3rd Qu.: 0.4332 3rd Qu.: 0.6024 3rd Qu.: 0.2683 3rd Qu.:3.000
## Max. : 0.4406 Max. : 3.5453 Max. : 2.9865 Max. :4.000
Original Boston dataset is now scaled and the result of K-means clustering is saved to the variable clust
Next the LDA is performed and the biplot with arrows is created
## Call:
## lda(clust ~ ., data = scaled_Boston)
##
## Prior probabilities of groups:
## 1 2 3 4
## 0.2114625 0.1304348 0.4308300 0.2272727
##
## Group means:
## crim zn indus chas nox rm
## 1 -0.3912182 1.2671159 -0.8754697 0.5739635 -0.7359091 0.9938426
## 2 1.4330759 -0.4872402 1.0689719 0.4435073 1.3439101 -0.7461469
## 3 -0.3894453 -0.2173896 -0.5212959 -0.2723291 -0.5203495 -0.1157814
## 4 0.2797949 -0.4872402 1.1892663 -0.2723291 0.8998296 -0.2770011
## age dis rad tax ptratio black
## 1 -0.6949417 0.7751031 -0.5965444 -0.6369476 -0.96586616 0.34190729
## 2 0.8575386 -0.9620552 1.2941816 1.2970210 0.42015742 -1.65562038
## 3 -0.3256000 0.3182404 -0.5741127 -0.6240070 0.02986213 0.34248644
## 4 0.7716696 -0.7723199 0.9006160 1.0311612 0.60093343 -0.01717546
## lstat medv
## 1 -0.8200275 1.11919598
## 2 1.1930953 -0.81904111
## 3 -0.2813666 -0.01314324
## 4 0.6116223 -0.54636549
##
## Coefficients of linear discriminants:
## LD1 LD2 LD3
## crim 0.18113078 -0.5012256 -0.60535205
## zn 0.43297497 -1.0486194 0.67406151
## indus 1.37753200 0.3016928 1.07034034
## chas -0.04307937 -0.7598229 -0.22448239
## nox 1.04674638 -0.3861005 -0.33268952
## rm -0.14912869 -0.1510367 0.67942589
## age -0.09897424 0.0523110 0.26285587
## dis 0.13139210 -0.1593367 -0.03487882
## rad 0.65824136 0.5189795 0.48145070
## tax 0.28903561 -0.5773959 0.10350513
## ptratio 0.22236843 0.1668597 -0.09181715
## black -0.42730704 0.5843973 0.89869354
## lstat 0.24320629 -0.6197780 -0.01119242
## medv 0.21961575 -0.9485829 -0.17065360
##
## Proportion of trace:
## LD1 LD2 LD3
## 0.7596 0.1768 0.0636
Biplot shows that variables indus, zn and medv are the most influencial separators for the clusters.
Colors of the both plots is based to four classes. It seems that K-means plot shows the different clusters more clearly than the plot that is based on the crime classification.
library(MASS)
library(ggplot2)
library(GGally)
library(corrplot)
library(dplyr)
library(plotly)
The data of this exercise originates from the United Nations Development Programme. Human development index (HDI) was created to emphasize that people and their capabilities should be the ultimate criteria for assessing the development of a country, not economic growth alone.
Original data can be found from: http://hdr.undp.org/en/content/human-development-index-hdi
Modified dataset includes 9 variables and 155 observations as follows:
| variable | description |
|---|---|
| country | Country name (name of the row) |
| gni_capita | Gross national income per capita |
| life_exp | Life expectancy at birth |
| edu_years | Expected years of schooling |
| mortality | Maternal mortality ratio |
| young_mom | Adolescent birth rate |
| women_parlament | Percentange of female representatives in parliament |
| edu_ratio | Ratio between females and males at least secondary education |
| labour_ratio | Ratio between females and males in the labour force |
## 'data.frame': 155 obs. of 8 variables:
## $ gni_capita : int 64992 42261 56431 44025 45435 43919 39568 52947 42155 32689 ...
## $ life_exp : num 81.6 82.4 83 80.2 81.6 80.9 80.9 79.1 82 81.8 ...
## $ edu_years : num 17.5 20.2 15.8 18.7 17.9 16.5 18.6 16.5 15.9 19.2 ...
## $ mortality : int 4 6 6 5 6 7 9 28 11 8 ...
## $ young_mom : num 7.8 12.1 1.9 5.1 6.2 3.8 8.2 31 14.5 25.3 ...
## $ women_parlament: num 39.6 30.5 28.5 38 36.9 36.9 19.9 19.4 28.2 31.4 ...
## $ edu_ratio : num 1.007 0.997 0.983 0.989 0.969 ...
## $ labour_ratio : num 0.891 0.819 0.825 0.884 0.829 ...
All the variables are either numerical or integers. Two of the variables are ratios (edu_ratio, labour_ratio).
## gni_capita life_exp edu_years mortality
## Min. : 581 Min. :49.00 Min. : 5.40 Min. : 1.0
## 1st Qu.: 4198 1st Qu.:66.30 1st Qu.:11.25 1st Qu.: 11.5
## Median : 12040 Median :74.20 Median :13.50 Median : 49.0
## Mean : 17628 Mean :71.65 Mean :13.18 Mean : 149.1
## 3rd Qu.: 24512 3rd Qu.:77.25 3rd Qu.:15.20 3rd Qu.: 190.0
## Max. :123124 Max. :83.50 Max. :20.20 Max. :1100.0
## young_mom women_parlament edu_ratio labour_ratio
## Min. : 0.60 Min. : 0.00 Min. :0.1717 Min. :0.1857
## 1st Qu.: 12.65 1st Qu.:12.40 1st Qu.:0.7264 1st Qu.:0.5984
## Median : 33.60 Median :19.30 Median :0.9375 Median :0.7535
## Mean : 47.16 Mean :20.91 Mean :0.8529 Mean :0.7074
## 3rd Qu.: 71.95 3rd Qu.:27.95 3rd Qu.:0.9968 3rd Qu.:0.8535
## Max. :204.80 Max. :57.50 Max. :1.4967 Max. :1.0380
There is greate variation between numeric variables. Variable like Gross national income per capita (gni_capita) differs between 581$ (min) -123,124$ (max) and at the same time there is two variables (edu_ratio, labour_ratio) that are ratios and differs both sides of number 1. Life expectancy (life_exp) and expected years of schooling (edu_years) are in years and the max observation is under 100 years. The percentange of female representatives in parliament (women_parlament) is the only variable that is represented in percentange.
Maternal mortality (mortality) ratio correlates strongly (negatively) with life expectancy, excpected years of schooling and education ratio between females and males. Maternal mortality and adolescent birth rates appears to be connected too. One could make conclusion that in some countries uneducated females give birth at very young age and that leeds to high maternal mortality rate.
Gross income per capita appears to be connected to life expectancy and expected years in schooling. Is it so that educated labour force helps society to prosper in economic sense.
Women in parliament seems to correlate with education rate between females and males. It seems that the gender equality starts from education and enables both genders to participate in social desicion-making.
First we conduct PCA with unstandardized data
## Standard deviations:
## [1] 1.854416e+04 1.855219e+02 2.518701e+01 1.145441e+01 3.766241e+00
## [6] 1.565912e+00 1.912052e-01 1.591112e-01
##
## Rotation:
## PC1 PC2 PC3 PC4
## gni_capita -9.999832e-01 -0.0057723054 5.156742e-04 -4.932889e-05
## life_exp -2.815823e-04 0.0283150248 -1.294971e-02 6.752684e-02
## edu_years -9.562910e-05 0.0075529759 -1.427664e-02 3.313505e-02
## mortality 5.655734e-03 -0.9916320120 -1.260302e-01 6.100534e-03
## young_mom 1.233961e-03 -0.1255502723 9.918113e-01 -5.301595e-03
## women_parlament -5.526460e-05 0.0032317269 7.398331e-03 9.971232e-01
## edu_ratio -5.607472e-06 0.0006713951 3.412027e-05 2.736326e-04
## labour_ratio 2.331945e-07 -0.0002819357 -5.302884e-04 4.692578e-03
## PC5 PC6 PC7 PC8
## gni_capita -0.0001135863 2.711698e-05 8.075191e-07 -1.176762e-06
## life_exp 0.9865644425 1.453515e-01 -5.380452e-03 2.281723e-03
## edu_years 0.1431180282 -9.882477e-01 3.826887e-02 7.776451e-03
## mortality 0.0266373214 -1.695203e-03 -1.355518e-04 8.371934e-04
## young_mom 0.0188618600 -1.273198e-02 8.641234e-05 -1.707885e-04
## women_parlament -0.0716401914 2.309896e-02 2.642548e-03 2.680113e-03
## edu_ratio -0.0022935252 -2.180183e-02 -6.998623e-01 7.139410e-01
## labour_ratio 0.0022190154 -3.264423e-02 -7.132267e-01 -7.001533e-01
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6 PC7
## Standard deviation 1.854e+04 185.5219 25.19 11.45 3.766 1.566 0.1912
## Proportion of Variance 9.999e-01 0.0001 0.00 0.00 0.000 0.000 0.0000
## Cumulative Proportion 9.999e-01 1.0000 1.00 1.00 1.000 1.000 1.0000
## PC8
## Standard deviation 0.1591
## Proportion of Variance 0.0000
## Cumulative Proportion 1.0000
The plot don’t look good. A large variance of the gross income per capita variable makes it too important for the principal component analysis. The data need to be standardized.
## gni_capita life_exp edu_years mortality
## Min. :-0.9193 Min. :-2.7188 Min. :-2.7378 Min. :-0.6992
## 1st Qu.:-0.7243 1st Qu.:-0.6425 1st Qu.:-0.6782 1st Qu.:-0.6496
## Median :-0.3013 Median : 0.3056 Median : 0.1140 Median :-0.4726
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.3712 3rd Qu.: 0.6717 3rd Qu.: 0.7126 3rd Qu.: 0.1932
## Max. : 5.6890 Max. : 1.4218 Max. : 2.4730 Max. : 4.4899
## young_mom women_parlament edu_ratio labour_ratio
## Min. :-1.1325 Min. :-1.8203 Min. :-2.8189 Min. :-2.6247
## 1st Qu.:-0.8394 1st Qu.:-0.7409 1st Qu.:-0.5233 1st Qu.:-0.5484
## Median :-0.3298 Median :-0.1403 Median : 0.3503 Median : 0.2316
## Mean : 0.0000 Mean : 0.0000 Mean : 0.0000 Mean : 0.0000
## 3rd Qu.: 0.6030 3rd Qu.: 0.6127 3rd Qu.: 0.5958 3rd Qu.: 0.7350
## Max. : 3.8344 Max. : 3.1850 Max. : 2.6646 Max. : 1.6632
## Standard deviations:
## [1] 2.0708380 1.1397204 0.8750485 0.7788630 0.6619563 0.5363061 0.4589994
## [8] 0.3222406
##
## Rotation:
## PC1 PC2 PC3 PC4 PC5
## gni_capita -0.35048295 -0.05060876 0.20168779 -0.72727675 0.4950306
## life_exp -0.44372240 0.02530473 -0.10991305 -0.05834819 -0.1628935
## edu_years -0.42766720 -0.13940571 0.07340270 -0.07020294 -0.1659678
## mortality 0.43697098 -0.14508727 0.12522539 -0.25170614 0.1800657
## young_mom 0.41126010 -0.07708468 -0.01968243 0.04986763 0.4672068
## women_parlament -0.08438558 -0.65136866 -0.72506309 0.01396293 0.1523699
## edu_ratio -0.35664370 -0.03796058 0.24223089 0.62678110 0.5983585
## labour_ratio 0.05457785 -0.72432726 0.58428770 0.06199424 -0.2625067
## PC6 PC7 PC8
## gni_capita -0.11120305 -0.13711838 -0.16961173
## life_exp 0.42242796 -0.43406432 0.62737008
## edu_years 0.38606919 0.77962966 -0.05415984
## mortality -0.17370039 0.35380306 0.72193946
## young_mom 0.76056557 -0.06897064 -0.14335186
## women_parlament -0.13749772 0.00568387 -0.02306476
## edu_ratio -0.17713316 0.05773644 0.16459453
## labour_ratio 0.03500707 -0.22729927 -0.07304568
## Importance of components:
## PC1 PC2 PC3 PC4 PC5 PC6
## Standard deviation 2.0708 1.1397 0.87505 0.77886 0.66196 0.53631
## Proportion of Variance 0.5361 0.1624 0.09571 0.07583 0.05477 0.03595
## Cumulative Proportion 0.5361 0.6984 0.79413 0.86996 0.92473 0.96069
## PC7 PC8
## Standard deviation 0.45900 0.32224
## Proportion of Variance 0.02634 0.01298
## Cumulative Proportion 0.98702 1.00000